Are scattering properties of graphs uniquely connected to their shapes ?

Are scattering properties Are scattering properties of graphs uniquely connected of graphs uniquely connected

to their shapes?to their shapes?Leszek Sirko, Oleh HulLeszek Sirko, Oleh Hul

Michał Ławniczak, Szymon BauchMichał Ławniczak, Szymon BauchInstitute of Physics

Polish Academy of Sciences, Warszawa, Poland

Adam Sawicki, Marek KuśAdam Sawicki, Marek KuśCenter for Theoretical Physics, Polish Academy of Sciences,Center for Theoretical Physics, Polish Academy of Sciences,

Warszawa, Poland

Trento, 26 July, 2012Trento, 26 July, 2012

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Can one hear the shape of a drumCan one hear the shape of a drum??

Is the spectrum of the Laplace operator unique on the planar domain with Dirichlet boundary conditions?

Is it possible to construct differently shaped drums which have the same eigenfrequency spectrum (isospectral drums)?

M. Kac, Can one hear the shape of a drum?, Am. Math. Mon. (1966)


OOnene can can’’tt hear the shape of a hear the shape of a ddrumrum C. Gordon, D. Webb, S. Wolpert, One can't hear the shape of a drum, Bull.

Am. Math. Soc. (1992)

C. Gordon, D. Webb, S. Wolpert, Isospectral plane domains and surfacesvia Riemannian orbifolds, Invent. Math. (1992)

T. Sunada, Riemannian coverings and isospectral manifolds, Ann. Math. (1985)


Isospectral drumsIsospectral drums

S.J. Chapman, Drums that sound the same, Am. Math. Mon. (1995)

Pairs of isospectral domains could be constructed by concatenating an elementray „building block” in two different prescribed ways to form two domains. A building block is joined to another by reflecting along the common boundary.


C. Gordon and D. Webb

TransplantationTransplantation

A

B

C

D

E

F G

A-B-G

A-D-F

B-E+F

D-E+G

-A-C-E

-B+C-D

C-F-G

For a pair of isospectral domains eigenfunctions corresponding to the same eigenvalue are related to each other by a transplantation


OOnene cannot cannot hear the shape of a hear the shape of a ddrumrum

Authors used thin microwave cavities shaped in the form of two different domains known to be isospectral.They checked experimentally that two billiards have the same spectrum and confirmed that two non-isometric transformations connect isospectral eigenfunction pairs.

S. Sridhar and A. Kudrolli, Experiments on not hearing the shape of drums, Phys. Rev. Lett. (1994)


Can one hear the shape of a drumCan one hear the shape of a drum??Isospectral drums could be distinguished by measuring their scattering poles

Y. Okada, et al., “Can one hear the shape of a drum?”: revisited, J. Phys. A: Math. Gen. (2005)


Quantum graphs and microwave Quantum graphs and microwave networksnetworks

What are quantum graphs?Scattering from quantum graphsMicrowave networksIsospectral quantum graphsScattering from isospectral graphsExperimental realization of isoscattering graphsExperimental and numerical resultsDiscussion


Quantum graphsQuantum graphsQuantum graphs were introduced to describe diamagnetic anisotropy in organic molecules:

Quantum graphs are excellent paradigms of quantum chaos:

In recent years quantum graphs have attracted much attention due to their applicability as physical models, and their interesting mathematical properties

T. Kottos and U. Smilansky, Phys. Rev. Lett. (1997)

L. Pauling, J. Chem. Phys. (1936)


A graph consists of A graph consists of nn verticesvertices (nodes) (nodes) connected by connected by BB bonds (bonds (edgesedges))On each bond of On each bond of aa graph graph the the one-dimensional one-dimensional Schrödinger equation is definedSchrödinger equation is defined

Topology is defined by nTopology is defined by nn connectivityn connectivity matrix matrix

The length The length matrixmatrix LLi,ji,j

Vertex scattering matrix Vertex scattering matrix ϭϭdefines boundary conditionsdefines boundary conditions

Quantum graphs, dQuantum graphs, definitionefinition

,

1, and are connected0, otherwisei j

i jC

22

, ,2 ( ) ( )i j i jd x k xdx

, ' , '

2ij j j j

iv

, ' , 'ij j j j

Neumann b. c. Dirichlet b. c.


Spectrum and wavefunctionsSpectrum and wavefunctions

,

( ) , ,, ,

ikL ji jU k ej j i mi j j m

, , ,( )

in ikx ou t ikx

i j i j i jx a e a e

( ), ', ',

'

out i inj jj ji j

ja a

det 0I U k ki

a U k a

Spectral properties of graphs can be written in terms of 2Bx2B bond scattering matrix U(k)

(1)

(3)

( 2 )

( 4 )

1,4L

1,2L1,3L

2,5L

2,6L1 2

5

6

4

3 ( 6 )

(5)


Scattering from graphsScattering from graphs

1 1

in ikx ou t ikxc e c e

2 2

in ikx ou t ikxc e c e

(1)

(3)

( 2 )

( 4 )

1,4L 1,2L

1,3L

2,5L

2,6L

1 2

5

6

4

3 ( 6 )

(5)


Microwave networksMicrowave networks

Microwave network (graph) consists of coaxial cables connected by joints

O. Hul et al., Phys. Rev. E (2004) Quantum graphs can be simulated by microwave networks


Hexagonal microwave networkHexagonal microwave network1

2

3

45

6

6 vertices15 bonds

nB


Equations for mEquations for microwave networkicrowave networkssContinuity equation for charge and current:Continuity equation for charge and current:

Potential difference:Potential difference:

1r

2r

, ,( , ) ( , )i j i jdq x t dJ x tdt dx

,,

( , )( , ) i j

i j

q x tV x t

C

, ,( , ) ( )i ti j i jq x t e q x

, ,( , ) ( )i ti j i jV x t e V x

0R


Equivalence of equationsEquivalence of equations

2 2

, ,2 2( ) ( ) 0i j i jd V x V xdx c

2, 2

,2

( )( ) 0i j

i j

d xk x

dx

, ,( ) ( )i j i jx V x 2

22kc

Current conservation:

Microwave networks Quantum graphs

Neumann b. c.

,, , , , 0

0i j

i j j i i j i jx L xj i j i

d dC V x C V xdx dx

Equations that describe microwave networks with R=0 are formally equivalent to these for quantum graphs with Neumann

boundary conditions


Can one hear the shape of a Can one hear the shape of a graph?graph?

One can hear the shape of the graph if the graph is simple and bonds lengths are non-commensurateAuthors showed an example of two isospectral graphs

B. Gutkin and U. Smilansky, Can one hear the shape of a graph?, J. Phys. A: Math. Gen. (2001)

baa

2a+2b

b

2a+ba+2b

a

b2a+3b

2a

ba

a+2b


Isospectral quantum Isospectral quantum graphgraphssR. Band, O. Parzanchevski, G. Ben-Shach, The isospectral fruits of representation theory: quantum graphs and drums, J. Phys. A (2009)Authors presented new method of construction of isospectral graphs and drums

b

c

2a

c b

D

N

N

D

N 2b

2c

aD

a

ND

N

D


Isoscattering quantum Isoscattering quantum graphgraphss

Authors presented examples of isoscattering graphs

Scattering matrices of those graphs are connected by transplantation relation

R. Band, A. Sawicki, U. Smilansky, Scattering from isospectral quantum graphs, J. Phys. A (2010)

b

c

2a

cb

D

N

N

D

N2b

2c

aD

a

1( ) ( ) , for II IS k T S k T k


IsoscatteringIsoscattering graphs, definitiongraphs, definition

Two graphs are isoscattering if their scattering phases coincide


)) ((Im log det ( ) Im log det ( )IIIS S

det ( ) iS A e

11 12

21 22

S SS S

S

Experimental set-up


Isoscattering microwave networksIsoscattering microwave networks

4 321a a

2c

2b

2a

c

c

b

b6 4

35

1

2

Two isoscattering microwave networks were constructed using microwave cables. Dirichlet boundary conditions were prepared by soldering of the internal and external leads. In the case of Neumann boundary conditions, vertices 1 and 2, internal and external leads of the cables were soldered together, respectively.

Network I Network II


Measurement of the scattering Measurement of the scattering matrixmatrix

2a

c

c

b

b6 4

35

1

2

4 321a a

2c

2b

S

11 12

21 22

S SS S


The scatteringThe scattering phase phase

)) ((Im log det ( ) Im log det ( )IIIS S

Two microwave networks are isoscattering if for all values of ν:


Importance of the scattering Importance of the scattering aamplitudemplitude

In the case of lossless quantum graphs the scattering matrix is unitary. For that reason only the scattering phase is of interest.

However, in the experiment we always have losses. In such a situation not only scattering phase, but the amplitude as well gives the insight into resonant structure of the system

det ( ) iS A e


( () )det ( ) det ( )II ISS


Scattering amplitudes and phases

Isoscattering networks

Networks with modifiedboundary conditions

O. Hul, M. Ławniczak, S. Bauch, A.Sawicki, M. Kuś, and L. Sirko,accepted to Phys. Rev. Lett. 2012

Transplantation relationTransplantation relation

32b

2c

a4 a1 2

b

c1

2a4

23

5 c 6

b

1( ( ))II ITS S T

1 11 1

T


SummarySummaryAre scatteringAre scattering propertiesproperties of graphs uniquely connected of graphs uniquely connected to their shapes? – to their shapes? – in general in general NONO!!

The concept of isoscattering The concept of isoscattering graphs graphs is not only theoretical is not only theoretical idea but could be also realized experimentallyidea but could be also realized experimentally

Scattering amplitudes and phases obtained from Scattering amplitudes and phases obtained from the the experiment are the same withinexperiment are the same within the the experimental errors experimental errors

Using transplantation relation it is possible to reconstruct Using transplantation relation it is possible to reconstruct the scattering matrix of each network using the scattering the scattering matrix of each network using the scattering matrix of the other onematrix of the other one


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Are scattering properties of graphs uniquely connected to their shapes ?